Differential Equations

First Order Equations

Linear Differential Equations of First Order

Definition of Linear Equation of First Order

A differential equation of type

\[y’ + a\left( x \right)y = f\left( x \right),\]

where \(a\left( x \right)\) and \(f\left( x \right)\) are continuous functions of \(x,\) is called a linear nonhomogeneous differential equation of first order. We consider two methods of solving linear differential equations of first order:

Method of Variation of a Constant

This method is similar to the previous approach. First it’s necessary to find the general solution of the homogeneous equation:

\[y’ + a\left( x \right)y = 0.\]

The general solution of the homogeneous equation contains a constant of integration \(C.\) We replace the constant \(C\) with a certain (still unknown) function \(C\left( x \right).\) By substituting this solution into the nonhomogeneous differential equation, we can determine the function \(C\left( x \right).\)

The described algorithm is called the method of variation of a constant. Of course, both methods lead to the same solution.

Initial Value Problem

If besides the differential equation, there is also an initial condition in the form of \(y\left( {{x_0}} \right) = {y_0},\) such a problem is called the initial value problem (IVP) or Cauchy problem.

A particular solution for an IVP does not contain the constant \(C,\) which is defined by substitution of the general solution into the initial condition \(y\left( {{x_0}} \right) = {y_0}.\)